Computing with Noise - Phase Transitions in Boolean Formulas

TL;DR

This paper explores how noisy logical gates in computing circuits affect their ability to accurately represent Boolean functions, using a statistical mechanics approach to identify phase transitions and performance bounds.

Contribution

It introduces a unified framework connecting information theory bounds with phase transitions in noisy Boolean circuits, enabling new insights into error rates and function complexity.

Findings

Bounds on circuit performance derived from information theory

Identification of phase transitions related to noise levels

Framework for analyzing error rates and function depth

Abstract

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. This framework paves the way for obtaining new results on error-rates, function-depth and sensitivity, and their dependence on the gate-type and noise model used.